If V be the volume of a tetrahedron and V¹ be the volume of another tetrahedron formed by the centroids of faces of the previous tetrahedron and V = KV¹ , then K is equal to

 Consider a tetrahedron with vertices O(0,0, 0), A(a,0, 0), B(0, b, 0) and C(0 , 0, c). 

$\text{ Volume }\mathrm{V}=\frac{1}{6}\left[\overrightarrow{\mathrm{a}}\overrightarrow{\mathrm{b}}\overrightarrow{\mathrm{c}}\right]$

Now centroids of the faces OAB, OAC, OBC and ABC arc G₁(a/3, b/3,0), G₂(3,0, c/3), G₃(0, b/3, c/3) and G₄(a/3, b/3, c/3), respectively.

${\mathrm{G}}_4{\mathrm{G}}_1=\overrightarrow{\mathrm{c}}/3,{\overrightarrow{{\mathrm{G}}_4\mathrm{G}}}_2=\overrightarrow{\mathrm{b}}/3,{\overrightarrow{{\mathrm{G}}_4\mathrm{G}}}_3=\overrightarrow{\mathrm{a}}/3$

Volume of tetrahedron by centroids
$\begin{array}{l}{\mathrm{V}}^{\mathrm{\prime }}=\frac{1}{6}\left[\begin{array}{lll}\frac{\mathrm{a}}{3}& \overrightarrow{\mathrm{b}}& \frac{\overrightarrow{\mathrm{c}}}{3}\end{array}\right]=\frac{1}{27}\mathrm{V}\\ \Rightarrow \mathrm{K}=27\end{array}$